Find the Remainder (x^4+8x^3+12x^2)/(x+1)

$\frac{x^{4} + 8 x^{3} + 12 x^{2}}{x + 1}$

Step 1

To calculate the remainder, first divide the polynomials.

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} + 7 x^{2}$

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$5 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$5 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 5 x$

$x^{3}$

$7 x^{2}$

$5 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$7 x^{2}$

$5 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$7 x^{2}$

$5 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$0$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$5 x$

$5$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$0$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$5 x$

$5$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$0$

$5 x$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 5$

$x^{3}$

$7 x^{2}$

$5 x$

$5$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$0$

$5 x$

$5$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$7 x^{2}$

$5 x$

$5$

$x$

$1$

$x^{4}$

$8 x^{3}$

$12 x^{2}$

$0 x$

$0$

$x^{4}$

$x^{3}$

$7 x^{3}$

$12 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$0$

$5 x$

$5$

The final answer is the quotient plus the remainder over the divisor.

$x^{3} + 7 x^{2} + 5 x - 5 + \frac{5}{x + 1}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$5$

Find the Remainder (x^4+8x^3+12x^2)/(x+1)

Related Questions

Support LCMGCF.com by disabling your adblocker.